1. Conduction electrons in a metal: the free-electron gas

1. Conduction electrons in a metal: the free-electron gas ≫ 1 From free atoms to solid state 2 From an electron gas to a free electron gas 3 The Drude model - a classical model for a free electron gas 4 The Sommerfeld theory - a quantum mechanical model of a free electron gas

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1.1 From free atoms to solid state

From free atoms to solid state. Consider a collection of free atoms. With the exception of the nucleus, these atoms consist of 1) inner, relatively tightly bound electrons, so called innershell electrons or core-electrons 2) outer, losely bound electrons, so called valence electrons. In this description, we can define the valence electrons as the electrons which orbit outisde those inner electrons which possess the electronic structure of a rare gas atom. For examples: Sodium : 1s2 2s2 2p6 3s1

→

Aluminium : 1s2 2s2 2p6 3s2 3p1

[Ne]3s1 →

Copper : 1s2 2s2 2p6 3s2 3p6 4s1 3d10

[Ne]3s2 3p1 →

[Ar]3d10 4s1

In aluminium, the 1s, 2s and 2p orbitals are completely filled, and so have the same electronic configuration as the rare gas atom Ne. This means that the attractive Coulomb-field from the nucleus is effectively screened, and why the three outer electrons (in the 3s and 3p orbitals) are more weakly bound than the inner-shell electrons. The same is true for the single valence electron in sodium and the eleven electrons in copper. What happens if we take the same neutral free atoms, e.g. aluminium, and give them the same separation they would have in the solid state, i.e., metallic aluminium? The inner-shell electrons are hardly affected. Conversely, the outer electrons - the valence electrons - can be said to move away from their respective atoms and move about like a more-or-less homogeneous electron gas. The atomic nuclei and the inner electrons, which remain in approximately unchanged atomic levels, form positive ions, i.e., ion cores. This description describes an important difference between free atoms and the solid state. An exception to this model are the nobel gases. By definition, these atoms do not have valence electrons, and in a rare-gas crystal, for example frozen Argon, the atomic electronic configuration is more-or-less unchanged. Valence electrons in the solid state or, to be more precise, their quantum mechanical wavefunction are only weakly or moderately localised around atoms; more importantly, they are not localised to any specific atom. In a single solid state crystal, the wavefunction of a valence electron is spread out over the entire crystal or, in a polycrystaline material, over the whole microcrystal. Strictly speaking, this is also true for those electrons in the RDT / FK5015 /

1.1 From free atoms to solid state

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inner shells, although their wavefunctions are vanishingly small in the areas between the ion cores, and so it is reasonable to consider these electrons as localised (bound) to specific ions. Another way of expressing this is to consider that the probability that these electrons can change location via tunneling from one atom to another is vanishingly small.

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2. Conduction electrons in a metal: the free-electron gas

2. Conduction electrons in a metal: the free-electron gas

1 From free atoms to solid state ≫ 2 From an electron gas to a free electron gas 3 The Drude model - a classical model for a free electron gas 4 The Sommerfeld theory - a quantum mechanical model of a free electron gas

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2.2 From an electron gas to a free electron gas

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From an electron gas to a free electron gas. With the help of band-gap theory, it can be seen that the electrons in a metal contribute to its electrical conductivity, and these are called the conduction electrons, though not every valence electron contributes. For example, metallic copper has 11 valance electrons but only one of them contributes to the copper’s conductivity. Consider a metal with n conduction electons per unit volume. These conduction electrons also represent a electrical charge density = −ne, where e is the elementary charge. The remaining valence electrons together with the inner shell electrons consitiute an average electrical charge denisty = +ne. Therefore, it is a reasonable approximation to assume that a single conduction electron sees its environment as being electrically neutral. With this assumption, the electron moves about without the influence of external forces, i.e., as a free particle. This model is called the free elctron gas model. It is noted that the model assumes that the positive charge density , +ne, is considered as a homogeneous continuum in which the conduction electrons move. It follows that the free electron gas, in equilibrium, is therefore also homogeneous.

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3. Conduction electrons in a metal: the free-electron gas

3. Conduction electrons in a metal: the free-electron gas

1 From free atoms to solid state 2 From an electron gas to a free electron gas ≫ 3 The Drude model - a classical model for a free electron gas 4 The Sommerfeld theory - a quantum mechanical model of a free electron gas

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3.3 The Drude model - a classical model for a free electron gas

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The Drude model - a classical model for a free electron gas. When Drude presented the first classical theory to describe a free electron gas in 1900, he assumed that the free moving electrons would follow MaxwellBoltzmann statistics, similarly to atoms in a classical ideal gas. Their average translational energy would then be 32 kB T where kB is Boltzmanns constant, and T the absolute temperature. Electric current in a metal means that the gas of conduction electrons is set in coordinated motion, e.g, via an external electric field. Electric resistance is dependent on the conduction electrons now and then colliding with something that changes their direction of motion. Drude supposed that the electrons scattered from the ion cores, but we now know that this is incorrect: the conduction electrons generally travel over distances that are considerably larger than the separation between ion cores without scattering, and this is described by band-gap theory. It has been found that the resistance in crystaline media is mostly due to the conduction electrons scattering off either vibrations in the crystal structure, i.e., thermal motion, or defects in the structure. The lattice vibrations are quantised, and these quanta are called phonons, and so the conduction electrons scatter by absorption and emission of phonons, i.e., via exchanging momentum (and a little energy) with the quantised lattice-vibration field. As a starting point, we can use the Drude model without specifing exactly what the conduction electrons are scattering off. Here, the Drude model is then an example of a phenomenological model. It should be noted that the free electron gas is an approximative model for conduction electrons, and that it does not work as a model for the other valence electrons in the metal or for the valence electrons in a crystaline isolator - these electrons do not contribute to the electrical conductivity. Electrical conductivity - Ohm’s law. Ohms law for the relationship between the current density, j, and the electric field strength, E can be expressed by: j = σE

(1)

where σ is the conductivity. We can prove this equation from Drudes model and a formula for σ. If we assume that every conduction electron occasionally scatters (against RDT / FK5015 /

3.3 The Drude model - a classical model for a free electron gas

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vibrations or defects) in such a way that after every collsion the electron moves in a random direction. The average time between two collisions (of the same electron) is called the relaxation time, τ . Furthermore, if we assume that we have n conduction electrons per unit volume in the metal, and that we set them in a uniform electic field, E. We assume that the electrons accelerate in this field according to Newton’s second law. Even in the presence of the electric field, the electrons have a definite terminal velocity, vth , but that their direction of travel is random and changes over time due to the scattering collisions. If E = 0, the average velocity vector of the electrons, i.e. their drift velocity, vd also zero. Assume that a collision can occur at any time with equal probability, which is reasonable if we assume that the metal is a homogeneous medium. In this case, the probability that the electron will not scatter under a time period, t, is given by exp(t/τ ). This can be understood in the following way. Assume that the probability of not undergoing a collision during a period of time t1 is P (t1 ), where P is some unspecified function. The probability of not underging a collision during two consecutive periods of time t1 and t2 must, from the law of the product of two independent probabilities, be given by P (t1 + t2 ) = P (t1 )× P (t2 ). The mathematical expression which satisfies this requirement is the exponential function, P (t) = exp(−t/τ ). This function gives the correct average time τ = 1 with t = 0 and decreases with increasing t, as it should. The simplest method to prove Ohm’s law is to start from a statistical equation of motion with the following form: dvd dt

= −

vd eE − τ m

(2)

We can motivate this equation in the following way. The right-hand side consists of two terms. The first of these represents the effect of relaxation, and the second the effect of the electric field. Next, assume that there are no scattering collisions with the media, i.e., no relaxation or resistance. We can achieve this by allowing τ to go to infinity, τ → ∞. Eqn 2 then reduces to: eE dvd = − (3) dt m which means that all the electrons accelerate together under the influence of the electric field, E. If we assume instead that at a given time, t = 0, the electric field is turned off, i.e., we set the field E = 0, then eqn 2 reduces to: dvd dt

= −

vd τ

(4)

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3.3 The Drude model - a classical model for a free electron gas

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The solution to this equation is: vd (t) = vd (0)e−t/τ

(5)

i.e., the drift velocity decreases exponentially with a relaxation time, τ . The current doesn’t stop instantaneously. Eqn. 5 is also in agreement with the probability P (t) = exp(−t/τ ) that an electron has not scattered after a time t. The explanation as to why the current decreases is not that the electrons go successively slower. In reality, the drift velocity vd is small compared to the individual electrons thermal average veleocity, vth . The current, i.e., the drift velocity vector vd decreases to zero because the conduction electrons through their collisions with the media lose memory of their direction of motion. Nevertheless, it is common though slightly incorrect, to occasionally refer to the second term on the right-hand side of eqn. 2 as a “friction” term. We can now prove Ohm’s law from eqn. 2. If we assume that we have static equilibrium, i.e. dvd /dt = 0, we obtain: vd = −

eEτ m

(6)

If we combine this expression with the known expression for electric current density: j = −nevd

(7)

we obtain the desired result: j = σE

(8)

Ohm’s law, where the conductivity, σ, is given by: σ =

ne2 τ m

(9)

Because the effect on the abolute value of the velocity of the applied electric field is small, we can say that, on average, the electrons move with the thermal velocity, vth , regardless of the electron field strength, E. The average distance the electrons travel betwen two scattering collisions is then: λ = τ · vth

(10)

which is usually called the conduction electrons free average wavelength or mean-free path. We should note that up until now have we assumed that in every collision the electron scatters in a completely random direction. In reality, the electron only scatters slighty in every collision (a glancing collision). However, a succession of such collisions has the same total effect RDT / FK5015 /

3.3 The Drude model - a classical model for a free electron gas

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as our assumed model single collision. The correct term for λ is the average electron transport length, as it describes the characteristic wavelength over which an electron, due to repeated collisions, loses the memory of its initial direction of motion. From eqn 10, the relaxation time, and, therefore, the conductivity, is given by λ/τ . Drude assumed that λ should be of the same order of magnitude as the separation between the atoms in the media, while vth should approximate the average velocity of a classical particle with a given temeprature. In reality, akthough the answer actually agrees quite well with empirical measurements of conductivity, these assumption are wrong: λ is considerably larger than the inter-atomic distance, typically 100× larger. Furthermore, as we will see, the velocity vth should be replaced with the so called Fermi velocity, vF , which is also typically 100× larger. However, this also clarifies why Drude’s classical model works so well: these two errors, each with a factor of 100, cancel each other out.

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4. Conduction electrons in a metal: the free-electron gas

4. Conduction electrons in a metal: the free-electron gas

1 From free atoms to solid state 2 From an electron gas to a free electron gas 3 The Drude model - a classical model for a free electron gas ≫ 4 The Sommerfeld theory - a quantum model of a free electron gas

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4.4 The Sommerfeld theory - a quantum model of a free electron gas

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The Sommerfeld theory - a quantum mechanical model of a free electron gas. The Sommerfeld model is the simplest quantum mechanical theory for a free electron gas. What differentiates it from the Drude model can be summarised quite briefly: using the Schr¨ odinger equation, a quantum mechanical deascription of the electrons in the free electron gas is obtained, taking into consideration the fact that an electron is a fermion, and must obey the Pauli exclusion principle, i.e., two or more electrons cannot occupy the same identical quantum state. Assume that there are N electrons contained in a cube of length L and volume V = L3 . The number of electrons per unit volume is then n = N/V . It is mathemtically convient to assume that the metal piece has the form of a cube, but it does not invalidate or limit the validity of the result we obtain. As before, assume that by “free” electron we mean that every electron moves unaffected by external forces, i.e., in a constant potential V . For simplicity, we can set V = 0. Furthertmore, we also ignore the electrostatic replusion between the electrons, as well as the electrostatic interactions between the conduction electrons and the non-conduction electrons and the atom nuclei. Relaxtion will be introduced later, and will have the same meaning as earlier, i.e., that occasionally every electron in the gas scatters into a new and random direction. In the assumptions we have made, because every electron in the gas moves around as if the other electrons do not exist, we can calculate the possible states using the Schr¨ odinger equation for a single free electron, i.e.: −

~2 ∇Ψ(r) = EΨ(r) 2m

(11)

The next step is to include the boundary conditions, i.e., the electrons are confined in a cube of length L. This is usually achieved by considering the solutions in the form of standing waves in the cube. However, instead we will use the so-called periodic boundary conditions. Motivation for the Periodic boundary conditions. One well known solution to eqn 11 is the plane wave: Ψk (r) =

1 √ eik·r V

(12)

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4.4 The Sommerfeld theory - a quantum model of a free electron gas

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which is already correctly normalised, such the the volume intergral of | Ψk | over the whole cube =1. Meanwhile, the plane wava also represents a free particle with an exactly defined momentum, ~k, but with a completely undefined position. The last point means that unless we know more we cannot accept eqn 12 as a solution because it means the electrons cannot be confined in the cube of volume V . Nevertheless, if we want to work with solutions of the form given in eqn 12 - as they do represent free particles - we must introduce an approximation. We assume that the electron has an completely defined momentum, despite being confined in the cube V , and due to Heisenberg’s uncertainty principle must have some uncertainty in its momentum. When is this approximation valid? If we consider the electron as a wavepacket, i.e. as a superposition of plane waves which have the form given by eqn 12, with a spatial distribution dv . The uncertaintly principle tells us that: δ(~ | k |) · dv ≈ ~,

(13)

where δ(~ | k |) is the uncertainty in the electrons momentum. If dv ≪ L we can ignore the effect on the electron from the surface of the cube. At the same time, δ(k) ≪ k so the electron has a well-defined momentum and can in a good approximation be described by a plane wave according to eqn 12. Another clear requirement that must be satisfied is λ ≪ L, where λ is the electrons de Broglie wavelength, λ = 2π/| k |. If this condition is not fulfilled, e.g. if λ ≈ L, the cube’s surfaces will have a measureable effect on the electrons state, and in this case the solutions should be considered as standing waves between these surfaces. Providing we do not forget that we have ignored the influence of the walls on the wavefunction, we can work with solutions having the form given by eqn. 12. The following now hold: the functions Ψk (r) = √1V eik·r where: 2π L 2π = ny L 2π = nz , L

kx = nx ky kz

(14)

where nx , ny , nz = 0, ±1, ±2, ±3 . . . define an orthogonal, normalised, complete basis set for other functions Ψ(x, y, z) defined in the cube, V . Any other function Ψ(x, y, z) defined from x, y, z within the cube, V , can always be written as a superposition of functions from 14. The superposition is, in itself, a Fourier transform of Ψ with a period L in both x−, y− and z− di-

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4.4 The Sommerfeld theory - a quantum model of a free electron gas

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rections. The states given in 14 are orthogonal means that they are different states with respect to the Pauli principle. It is usual to state that the functions or (kx , ky , kz )-values in 14 are calculated with periodic boundary conditions. This means that, for example, kx shall be chosen so that the function exp(ikx x) is periodic with a period L so that: exp(ikx (x + L)) = exp(ikx x)

(15)

It is easy to show that this leads to 14. What this demonstrates is the key idea to the periodic boundary conditions: they happen to generate a complete set of plane-wave solutions to the Schr¨ odinger equation which are othogonal within the cube V . The ground state of the free electron gas. In k−space, a cartesian coordinate system with the axes (kx , ky , kz ), we can form k-vectors given in eqn. 14, i.e., those allowed according to the periodic boundary conditions, forming a lattice of points. This is shown in Fig. 1.

Figure 1: The Fermi sphere The number of points per unit volume in k-space is (L/2π)3 , i.e., if we consider that every point in the lattice lies in the centre of its own little cube of volume (2π/L)3 . Every point represents two electrons according to the Pauli exclusion principle: one with spin up (~/2) and one with spin down (−~/2). The energy of this state is given by: E =

~2 2 k 2m

=

~2 kx2 + ky2 + kz2 2m

(16)

In the ground state, i.e., where the total energy of the gas is as low as possible, all of the N electrons will be found in the state with the lowest RDT / FK5015 /

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4.4 The Sommerfeld theory - a quantum model of a free electron gas

possible energy, i.e., as close to the origin (k = 0) as possible, taking into consideration the exclusion principle which allows two electrons in each and every point in the lattice. If N is extremely large (≫ 1), then the filled states will form a spherical volume in k−space with its centre at the origin. This is the Fermi sphere (Fig. 1). If the volume was not spherical, an electron at the surface could move to a aite closer to the centre and, thus, be in a lower energy state. The radius of this sphere is called the Fermi wave-vector, kF , and the associated energy of an electron in a state which lies on the surface of the Fermi sphere (the Fermi surface) is the Fermi energy, EF , and is given by: EF

=

~2 kF2 2m

(17)

A free electron gas is an example of a Fermi gas, i.e. a gas of fermions. The Fermi wave-vector can be obtained by noting that the volume of the Fermi sphere mupltipled by the the number of states per unit volume (in k-space) must equal the number of electrons, N : " 3 # 4 3 L πkF · 2 N = (18) 3 2π and rearranging: kF

= (3πn)1/3

(19)

where n = N/V which, as earlier, is the number of electrons per unit volume (in normal space). Thus: EF

=

2/3 ~2 3π 2 n 2m

which states that the Fermi energy is larger of the electron gas is denser. The so-calledd Fermi temperature is a fictional temperature defined as EF = kB TF . Finally, we obtain the Fermi velocity of an electron on the surface of the Fermi sphere: 1/3 ~ ~kF = 2π 2 n vF = m m In practice, the Fermi energy in metals is on the order 1-10 eV. This means that the average kinetic energy of a conduction electron in a metal is considerably larger than would be expected from classical theory. Electrical conductivity. How are free electrons influenced by an electric field, E? RDT / FK5015 /

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4.4 The Sommerfeld theory - a quantum model of a free electron gas

For an electron in an allowed state 12 the momentum is p = ~k. Assume that we can use Newton’s Second Law. Here, dp/dt = −eE, so: ~

d k = −eE dt

(20)

In other words, an electron in a plane-wave state Ψk (r) at a time t should at a time t + dt find itself in a plane-wave state Ψ′k (r), where: eE ′ dt (21) k = k + dk = k − h This is quantum mechanically exact even if we used a classical argument. The effect of the electric field, E, can be therefore seen as that all electrons in the Fermi sphere at the same time move to a new state in accordance with eqn. 20: naturally this is not against the exclusion principle. Furthermore, the result is that the Fermi sphere translates in k−space with a velocity dk/dt = −eE/~. The process is illustrated in Fig. 2.

Figure 2: The offset Fermi sphere If we denote the wavenumber vector at the centre of the Fermi sphere as km (Fig. 2), we can clearly describe the effect of the field E with the equation: ~

d km = −eE dt

(22)

As in the Drude model, the resistance depends on the fact that electrons scatter against the vibrations and defects in the crystal structure. With these scattering processes, the changes in the energy E of the electrons is generally significantly smaller than the Fermi energy, EF . Consequently, these scattering processes can only take place for those electrons which are RDT / FK5015 /

4.4 The Sommerfeld theory - a quantum model of a free electron gas

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located on or close to the Fermi surface. It is noted that an electron which is located deeper in the Fermi sphere must therefore absorb considerably more energy to move to a free state, i.e., a state wich is either on or outside the Fermi surface. As earlier, λ is the mean free wavelength (more exactly, average electron transport wavelength) between the scattering collisions. Becasue the scattering processes only involve electrons in states close to the Fermi surface, we take the relaxation time to be: λ (23) τ = vF Compare this with eqn.10. An important point is that the Fermi velocity vF is significantly larger than the classical average thermal velocity vth . In the so-called relaxation approximation, the effect of the scattering process is descibed such that the Fermi sphere returns to it’s relaxed state (km = 0) according to the equation: 1 d km = − kb dt τ

(24)

(compare with eqn. 4). The combination of the effect of the electric field and the relaxation time is given by (compare eqn. 2): d e 1 km = − E − kb dt ~ τ

(25)

For a stationary state, i.e., constant current, the time derivative is zero and: e ~km = − E (26) ~ We can now calculate the electric current density, j, from eqn. 26. As the average value of the electron’s wave vector is just km , we determine that the aveage value of the group velocity, i.e. the drift velocity d is: vd =

~km m

(27)

Using eqns 27 and 26 we can see that the electrical current density is: j = −nevd

=

ne2 τ E m

=

σE

(28)

And we obtain the following formula for the conductivity: σ =

ne2 τ E m

=

σ

ne2 λ E mvF

(29)

which is very similar to that given by the Drude model eqn 9. RDT / FK5015 /

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Matthiessen’s Rule. As we noted earlier, the relaxation is mostly dependent on two processes: scattering against the vibrations in the crystal structure and scattering against different kinds of defects in the crystal structure. These two processes can be seen as approximately independent. Denote the average electron transport wavelength for the first of these processes is λg and that for the second is λd . As the vibrations arise from the crystal structure’s thermal motion, it is reasonable to assume that the scattering increases with increasing temperature, i.e., λg decreases. In the second process, λd , must depend to a large extent on the density of defects in the crystal and to a first approximation is independent of the temperature (true for the most part). These approximations form Matthiessen’s Rule.

Figure 3: Resitivity vs temperature We can then associate a relaxation time, τg = λg /vF , for scattering against vibrations and second relaxation time, τd = λd /vF , for scattering off defects. If the processes are independent, the total resistivity should be the sum of the resistivities (the inverse of the conductivities) related to these processes, i.e.: m 1 m 1 m = + 2 + (30) ρ = ρg + ρd = ne2 τg ne τd ne2 τg τd If we compare this with the formula for conductivity eqn 29, we see that the total relaxation time is given by: 1 1 1 = (31) + τ τg τd

A schematic picture of a typical temperature dependence of the resistivity of a metal is given in Fig. 3. At low temperatures, the contribution from scattering from the defects dominates. At higher temperatures, the contribution from scattering off the vibrations in the structure dominates - the resistivity tends to be proportional to the temperature. RDT / FK5015 /